The Double Root

When the discriminant is zero, the equation has one solution. Take x^2 - 4x + 4 = 0. The left side is a perfect square trinomial, so you can write it as (x - 2)^2 = 0. A square is zero only when the number being squared is zero, so x - 2 = 0 and x = 2. There is one solution.

You can also factor the left side as (x - 2)(x - 2) = 0. Both factors give the same equation, x - 2 = 0, so both give the same solution. A solution that comes from two equal factors is called a double root. The double root of this equation is 2.

The discriminant of x^2 - 4x + 4 = 0 is also zero. Here b^2 is 16 and 4ac is 16, so the discriminant is 16 - 16 = 0. Because it is zero, the plus and the minus in the quadratic formula give the same value, x = 2. The formula, the factoring and the squared form all agree that the equation has one solution.